Question

A thermal neutron with a speed v corresponding to the average thermal energy at
temperature T = 300 K is incident on a crystal. Will a diffraction pattern be
observed? Explain.

Answer 1

$$\dfrac{1}{2}mv^2=E=k_B T \Rightarrow v=\sqrt{\dfrac{2k_B T}{m}}$$

$$\lambda=\dfrac{h}{mv}=\dfrac{h}{\sqrt{2k_B Tm}}$$

$$\lambda=\dfrac{6.626 \cdot 10^{-34}}{\sqrt{2\cdot 300 \cdot 1.38 \cdot 10^{-23} \cdot 1.674 \cdot 10^{-27}}} \approx 1.78 \cdot 10^{-10}\, m$$

If the wavelength $\lambda$ is on the same order as the atomic separations, then neutrons will diffract when passing through the crystal.